The Automorphism Group of a Compact Group Action

This paper contains results on the structure of the group, DiffG(M), of equivariant C-diffeomorphisms of a free action of the compact Lie group G on M. DiffJjíAÍ) is shown to be a locally trivial principal bundle over a submanifold of T>\tf(X), X the orbit manifold. The structural group of this bundle is lf(G, M), the set of equivariant C'-diffeomorphisms which induce the identity on X. E?(G, M) is shown to be a submanifold of Diff^Af) and in fact a Banach Lie group (r < ~). 0. Introduction. This paper studies the group of equivariant diffeomorphisms of a smooth action of a compact Lie group G on a compact manifold M. Specifically most of the paper deals with the case when G acts freely on M. In this case there is an orbit manifold X, and an equivariant C-diffeomorphism / of M induces a C-diffeomorphism / of X. This defines a homomorphism P: DiffG(Af) —► Diff(X) (DiffG(A/) is the group of equivariant diffeomorphisms on M of class C, 1 < r < °°). We obtain some results about the structure of P. We show P admits smooth local cross-sections (Theorem 3.5). The kernel of P is the group of C-equivariant diffeomorphisms of M which induce the identity on X. This group is the structural group of the locally trivial principal bundle determined by P. We show ker F is a smooth submanifold of Diff(A/) and, that with respect to the induced differential structure, ker P is a Banach Lie group (Theorem 4.2). Recall Diff(M) is not a Banach Lie group as composition is not C1 (r < °°). The main technique introduced is the construction in §2 of G-lifts of sprays on X. This allows a precise connection between the manifold of maps differential structures on DiffG(7W) and Diff(X). 1. Preliminaries. M will always denote a compact, connected, C°°-manifold, G a compact Lie group. Assume G acts on M (on the left). We denote by X the orbit space and we let n: M —> X be the orbit projection. If G acts freely and differentiably (C°°) then X has a natural C°° -structure such that tt Presented to the Society, January 26, 1973 under the title The automorphism group of a compact Lie group action; received by the editors March 22, 1973. AMS (A/OS) subject classifications (1970). Primary 58D05, 22E65; Secondary 55F10.